Entanglement verification of noisy NOON states

Entangled quantum states, such as NOON states, are of major importance for quantum technologies due to their quantum-enhanced performance. At the same time, their quantum correlations are relatively vulnerable when they are subjected to imperfections. Therefore, it is crucial to determine under which circumstances their distinct quantum features can be exploited. In this paper, we study the entanglement property of noisy NOON states. This class of states is a generalization of NOON states including various attenuation effects, such as mixing, constant or fluctuating losses, and dephasing. To verify their entanglement, we pursue two strategies: detection-based entanglement witnesses and entanglement quasiprobabilities. Both methods result from our solution of so-called separability eigenvalue equations. In particular, the entanglement quasiprobabilities allow for a full entanglement characterization. As examples of our general treatment, the cases of NOON states subjected to Gaussian dephasing and fluctuating atmospheric losses are explicitly studied. In any correlated fluctuating loss channel, entanglement is found to survive for nonzero transmissivity. In addition, an extension of our approach to multipartite systems is given, and the relation to the quantum-optical nonclassicality in phase space is discussed.

Figure 1

Entanglement quasiprobability distribution of pure NOON states. A patch in the horizontal plane corresponds to a product $|a,b\rangle$ for the given vectors at the axes. Note, the solutions are symmetric with respect to subsystems in the considered case. The height of the corresponding bar is the value of $P_{\mathrm{Ent}}(a,b)$ in Eq. (28). The negativities directly reveal the entanglement of the NOON state.

Figure 2

Entanglement conditions are shown as functions of the strength of Gaussian dephasing $\delta$ for a dephased NOON state with $N=2$. Negative values indicate the entanglement detection according to the witness condition, Eq. (30) (dashed), and the negativity of the entanglement quasiprobabilities in Eq. (31) (solid).

Figure 3

Entanglement of a NOON state under correlated fluctuating loss, characterized via the fourth-order moment $\langle T^4\rangle$ of the probability distribution of the transmission coefficient $T$. Negative values certify entanglement. The dashed and dotted lines correspond to the witness conditions in Eqs. (33) and (35), respectively. The solid line shows the negativity of the quasiprobability in Eq. (37).

Figure 4

The expectation value of the witness (39) for partial (solid line) and full (dashed line) separability of a tripartite NOON state with dephasing of strength $\delta$ in one mode. A negative value certifies the presence of the corresponding form of inseparability.